Simplify the following expression: $z = \dfrac{54a^2 + 63a}{-90a^2 + 72a}$ You can assume $a \neq 0$.
Find the greatest common factor of the numerator and denominator. The numerator can be factored: $54a^2 + 63a = (2\cdot3\cdot3\cdot3 \cdot a \cdot a) + (3\cdot3\cdot7 \cdot a)$ The denominator can be factored: $-90a^2 + 72a = - (2\cdot3\cdot3\cdot5 \cdot a \cdot a) + (2\cdot2\cdot2\cdot3\cdot3 \cdot a)$ The greatest common factor of all the terms is $9a$ Factoring out $9a$ gives us: $z = \dfrac{(9a)(6a + 7)}{(9a)(-10a + 8)}$ Dividing both the numerator and denominator by $9a$ gives: $z = \dfrac{6a + 7}{-10a + 8}$